This is called the bulk-scattering correspondence, and is proved in [1]}. The proof goes as follows. Pick a section \(\widehat{\psi }_{\mathrm {in}}\)  on \(U_\mathrm {in} \cap \lbrace \kappa <0\rbrace \) . The boundary condition required in the definition of the scattering states naturally defines a scattering map \(\mathcal {S} : \mathcal {E}_{k_x,-\kappa } \rightarrow \mathcal {E}_{k_x,\kappa }\)  where \(\mathcal {E}_{k_x,\kappa }\)  is the fiber above \((k_x,\kappa )\) . Thus one naturally has the abstract section \(\psi _{\mathrm {out}}(k_x,\kappa ) = \mathcal {S} \widehat{\psi }_{\mathrm {in}}(k_x,-\kappa )\) . Then, on \(U_\mathrm {in} \cap U_\mathrm {out}\)  one may use the same section for in and out, namely \(\widehat{\psi }_{\mathrm {in}} = \widehat{\psi }_{\mathrm {out}}\)  so that we have \(\psi _{\mathrm {out}}(k_x,\kappa ) = S(k_x,\kappa ,a) \widehat{\psi }_{\mathrm {in}}(k_x,-\kappa )\) . Thus \(S\)  appears as a transition function between in and out section. Consequently, if \(U_\mathrm {in} \)  and \(U_\mathrm {out}\)  cover \(S^2\)  and if the loop \(\Gamma _{\delta ,a_0}\)  splits \(S^2\)  with interior contained in \(U_\mathrm {in}\)  and exterior contained in \(U_\mathrm {out}\) , then the winding of \(S\)  along it is the Chern number. One can check that this is the case with the choice (REF ) for \(S\)  as long as \(\zeta \)  is inside \(\Gamma _{\delta ,a_0}\) .
